Sensor Placement Estimation Force Based and Deflection Based Models

2/14 Deflection Estimation using Beam Theory Curvature (1/ρ) Slope Deflection y x x x dy dx d2yd2y dx 2 εxεx d Curvature = 1 ρ = x2x2 x1x1 f(x) = ax 2 +bx+c Sensor 1 Sensor 2 ε x : strain measured by FBG sensor ρ: radius of curvature d: distance from neutral axis Slope = ∫ f(x) dx Deflection = ∫∫ f(x) dx

Force-Based Displacement 2 locations of force concentration defined. Moment of bending calculations determine curvature. True deflection model calculated from curvature. Estimate curvature determined from x1, x2, two sensor locations. Estimated deflection = ∫∫curvature.

Model Construction F1F1 F2F2 Sensor 1 Sensor 2 x1x1 x2x2 2 / L L = 15cm Tip Deflection

Sensitivity The sensitivity of the positions x1 and x2 where found as the deflection error at the tip of the needle with respect to x1 and x2.

Results: Force-Based Model Magnitudes of forces had no effects on distribution of deflection error and sensitivities. Whether the forces where applied in opposing directions also had no effect. Only location of the applied forces changed the distribution of deflection error and sensitivities.

Error Distribution 2 Cases of varying loads applied at the same two locations. One case also had opposing forces.

Error Distribution 2 Cases of loads applied at differing locations.

Error & Sensitivities Distributions Force of 1 gram at half the needle length (75mm) and 2 grams at the tip (150mm). Deflections Sensitivity to x1 Sensitivity to x2

Deflection & Curvature Estimate Force of 1 gram at half the needle length (75mm) and 2 grams at the tip (150mm). The sensors are placed at 25mm and 82mm (based on sensitivity and deflection error) results.

Deflection Based Model Given a known curve, 3 rd order equations for each deflected segment (2) are found. (xd, yd) (xd2, yd2) L y(0) = 0 θ(0) = 0

BC’s and System of Equations The system of equations to solve is: Because the deflection and slope at x = 0 is zero, the first equation has no first order or y-intercept terms. The continuity constraints are:

Finding Coefficients With at least the point of deflection and a point in the y2 region, the system can be solved. A closer solution is found with more (x,y) points.

Deflection Based Method Equations for Y1 and Y2 are found. A true curvature model is found by taking the second derivative of the deflection. An estimate curvature is calculated from the actual curvatures at two sensor positions. Estimated deflection = ∫∫curvature.